`NIntegrate[x^50*Sin[x], {x, 0, 1}]` works fine and results to `0.0162898`.

Also putting `N[Integrate[x^50*Sin[x], {x, 0, 1}], 20]` you get a correct result.

If you try to calculate the anti-derivative using :

`anti[x_] := Module[{t}, Integrate[t^50*Sin[t], t] /. t -> x]` then you will see that you get the exact integral form.

Now 

`anti[1]==16432804687774250383441481995831940788236063969597816674967907249 Cos[1] + 25592576958484906358990554067594971071796795711786890514692693650 Sin[1]` 

and 

`anti[0]==30414093201713378043612608166064768844377641568960512000000000000`

`N[anti[1] - anti[0], 20]==0.016289783620195801683` but `N[anti[1]-anti[0]]==0.` which is caused by the round to only 8 digits of `Sin[1]` and `Cos[1]`.

Try `Trace[N[Integrate[x^50*Sin[x], {x, 0, 1}]]]` to see where exactly it goes way off!